Territory covered by N random walkers on fractal media: The Sierpinski gasket and the percolation aggregate - art. no. 011105

We address the problem of evaluating the number S-N(t) of distinct sites vi sited up to time t by N noninteracting random walkers all starting from the same origin in fractal media. For a wide class of fractals (of which the p ercolation cluster at criticality and the Sierpinski gasket are typical exa mples) we propose, for large N and after the short-time compact regime, an asymptotic series for SN(t) analogous to that found for Euclidean media: S- N(t) approximate to (S) over cap (N)(t)(1 - Delta). Here (S) over cap (N)(t ) is the number of sites (volume) inside a hypersphere of radius L[ln(N)/c] 1/upsilon where L is the root-mean-square chemical displacement of a single random walker, and upsilon and c determine how fast 1 - Gamma (t)(l) (the probability that a given site at chemical distance l from the origin is vis ited by a single random walker by time t) decays for large values of l/L: 1 - Gammat(l)similar to exp[-c(l/L)(upsilon)]. For the fractals considered i n this paper, upsilon = d(w)(l)/(d(w)(l) - 1), d(w)(l) being the chemical-d iffusion exponent. The corrective term Delta is expressed as a series in ln (-n)(N)ln(m) ln(N) (with n greater than or equal to 1 and 0 less than or eq ual to m less than or equal to n), which is given explicitly up to n = 2. T his corrective term contributes substantially to the final value of S-N(t) even for relatively large values of N.